A Lie algebra is the infinitesimal approximation to a Lie group.
A Lie algebra is a vector space $\mathfrak{g}$ equipped with a bilinear skew-symmetric map $[-,-] : \mathfrak{g} \wedge \mathfrak{g} \to \mathfrak{g}$ which satisfies the Jacobi identity:
A homomorphism of Lie algebras is a linear map $\phi : \mathfrak{g} \to \mathfrak{h}$ such that for all $x,y \in \mathfrak{g}$ we have
This defines the category LieAlg of Lie algebras.
The notion of Lie algebra may be formulated internal to any linear category. This general definition subsumes as special case generalizations such as super Lie algebras.
Given a commutative unital ring $k$, and a (strict for simplicity) symmetric monoidal $k$-linear category $(C,\otimes,1)$ with the symmetry $\tau$, a Lie algebra in $(C,\otimes,1,\tau)$ is an object $L$ in $C$ together with a morphism $[,]: A\otimes A\to A$ such that the Jacobi identity
and antisymmetry
hold. If $k$ is the ring $\mathbb{Z}$ of integers, then we say (internal) Lie ring, and if $k$ is a field and $C=Vec$ then we say a Lie $k$-algebra. Other interesting cases are super-Lie algebras, which are the Lie algebras in the symmetric monoidal category $\mathbb{Z}_2-Vec$ of supervector spaces and the Lie algebras in the Loday-Pirashvili tensor category of linear maps.
Alternatively, Lie algebras are the algebras over certain quadratic operad, called the Lie operad, which is the Koszul dual of the commutative algebra operad.
Lie algebras are equivalently groups in “infinitesimal geometry”.
For instance in synthetic differential geometry then a Lie algebra of a Lie group is just the first-order infinitesimal neighbourhood of the unit element (e.g. Kock 09, section 6).
More generally in geometric homotopy theory, Lie algebras, being 0-truncated L-∞ algebras are equivalently “infinitesimal ∞-group geometric ∞-stacks” (e.g. here), also called formal moduli problems (see there for more).
Notions of Lie algebras with extra stuff, structure, property includes
extra property
extra structure
extra stuff
See
Lie algebra
Examples of sequences of local structures
geometry | point | first order infinitesimal | $\subset$ | formal = arbitrary order infinitesimal | $\subset$ | local = stalkwise | $\subset$ | finite |
---|---|---|---|---|---|---|---|---|
$\leftarrow$ differentiation | integration $\to$ | |||||||
smooth functions | derivative | Taylor series | germ | smooth function | ||||
curve (path) | tangent vector | jet | germ of curve | curve | ||||
smooth space | infinitesimal neighbourhood | formal neighbourhood | germ of a space | open neighbourhood | ||||
function algebra | square-0 ring extension | nilpotent ring extension/formal completion | ring extension | |||||
arithmetic geometry | $\mathbb{F}_p$ finite field | $\mathbb{Z}_p$ p-adic integers | $\mathbb{Z}_{(p)}$ localization at (p) | $\mathbb{Z}$ integers | ||||
Lie theory | Lie algebra | formal group | local Lie group | Lie group | ||||
symplectic geometry | Poisson manifold | formal deformation quantization | local strict deformation quantization | strict deformation quantization |
Discussion with a view towards Chern-Weil theory is in chapter IV in vol III of
Discussion in synthetic differential geometry is in